Quantum incremental movement can be considered to be matter popping in and out of 3D spatial existence. It might seem like there is no purely mathematical reason why such an increment needs to be some smallest possible amount, e.g., the Planck length.
However, there are indeed purely logical reasons why there must be a smallest possible increment...
To claim potential infinite divisibility, is to assert that it is possible to divide space an infinite number of times.
Dividing any finite distance an infinite number of times will unequivocally result in the numerical value 0; infinitesimal? Don’t be silly.
Travelling 0 distance any number of times, or indeed infinitely many times, results in 0 distance travelled.
This is a contradiction, because if something is divided such that it no longer exists (hence 0), then all its meaning has been lost... consider:
Space itself has been eliminated from existence because of asserting infinite divisibility.
Regardless of space no longer existing, if say, a particle popped out of existence, and then popped back into existence at the exact (hence 0 again) same position, no movement will have occurred.
Pure logic applies to the idea of infinity – not simply the axiom of infinity or any other such assumptions, e.g., the assertion that convergent series (asymptotes) represent the possibility of being able to travel infinitely many discrete distances – an impossibility of course.
Description: The popular solution attempts to explain the possibility of movement by simply identifying the perfectly logical correspondence between infinitely many divisions ever decreasing in size and infinitely many smaller periods of time – crossing over infinitely many discrete distances within any given finite distance, e.g., 1/16 + 1/8 + 1/4 + 1/2, ad infinitum.
The video discusses the paradox itself, not the popular solution mentioned above.
Observation: Divisions of distance and time need to each have values greater than 0 to exist. Therefore, an infinite number of divisions will result in both an infinite distance and an infinite period of time. This paradox is simply a combination of those described in section: “Infinitesimal time & distance”
A quick look at calculus – Zeno's “Divisions of Motion”: Limitations of Calculus – Asymptotic Relationships
Description: In a race the quickest runner can never overtake the slowest runner because the faster runner must always reach the slower runner's previous position.
Cheating: The dichotomy paradoxical logic whereby the possibility of movement is challenged via infinite divisibility, is effectively being applied to the faster runner, but not to the slower runner.
Time as considered here is simply relative position change – consider the following:
B and C are each incrementing 1 unit of distance (1 “quantum”), i.e., the width of 1 square in this case.
The time that has passed from the perspective of any given square is the measurement of change in its position relative to the position of any other square.
Therefore, each time step (i.e., quanta of movement) exists as: “A” seeing B and C to each be moving at a rate of 1 square... “B” sees A moving at a rate of 1, and C at a rate of 2... and “C” sees A moving at a rate of 1, and B at a rate of 2.
Conclusion: 1 increment step is represented by 1 square, i.e., fixed size quanta – therefore the individual B passes the individual C at a rate of 2 quanta (1 quantum by B, + 1 quantum by C), per 1 quantum of B passing A.
The explanation for the moving arrow is basically the same as described above for Zeno's “Moving Rows”, but a more detailed example is below:
(The example below isn’t a special relativity example of course)
Let’s consider that there exists in the universe, nothing more than just one arrow, and lots of blocks like those described in the moving rows example above.
Arrow’s starting position: X=1, Y=1, Z=1 (1 unit is simply the width of 1 block).
Construct a clock by imagining the blocks to all be moving at different speeds relative to one another, i.e., imagine all their relative rates of movement to be a reference for judging how fast the arrow is moving, and indeed themselves.
The arrow pops out of existence and reappears at position: X=1, Y=1, Z=9.
Notice that “Z” has changed from 1 to 9.
The arrow moved 9 units directly towards one particular moving block “A”.
Two of the other moving blocks have each moved directly towards block A along the same axis, i.e., both blocks are simply moving inline – block “B” moved 2 units; block “C” moved 6 units.
Hypothetical inline-type clock – represented by the distance between B and C.
Block A defined speed to be: 1 mile per minute = 7 units of distance travelled closer to its own position... per change of 9 units between blocks B and C.
Clock = 6 – 2 = 4 units... so, 4 ÷ 9 = 0.44
B = (2 ÷ 7) ÷ 0.44 = 0.65 miles per minute.
C = (6 ÷ 7) ÷ 0.44 = 1.95 miles per minute.
Arrow = (9 ÷ 7) ÷ 0.44 = 2.92 miles per minute.
Notice: if C moves faster, the clock’s value is larger, which results in slower velocities being recorded... so a faster running clock makes everything seem slower, as expected of course.
Conclusion: time, movement, and therefore velocity and acceleration, are nothing more than relative changes in position... logically speaking that is. Causality and time are effectively the same thing. Causality can be thought of as the physical laws of existence in action – either changing the position of particles, or causing a position in space to become occupied via a particle popping into our 3D spatial reality.
A more detailed look at time – nothing more than change: “Time Steps vs Causality vs Block Universe”
Particles can be considered to exist as solid objects, pure energy, probabilistic waves, or field excitations (QFT) – the most intuitive being solid objects because that's how we experience them consciously.
According to quantum mechanics, spacetime itself is foaming with virtual particles that pop in and out of existence on the threshold of 3D spatial reality. The universe is considered to exist as indivisible chunks (quanta), like pixels on a computer screen.
Consider what movement and time are if only 1 single particle exists in the 3D spatial existence (assumed to pre-exist independent of physical matter) of the universe – like having the ability to light up just 1 single pixel on a computer screen at any 1 position.
Imagine there's no quantum foaming or anything else in existence; just this 1 hypothetical pixel, assumed to be timeless within itself, and that its hardware is non-existent... for argument's sake of course... just 1 timeless pixel.
The pixel exists at position X=5, Y=5, of possible screen coordinates 0 to 100 in both X and Y.
Position X=5 now increments by 1 until it reaches the edge of the screen at X=100.
Now remember that nothing else exists for making a comparison of its existence at different positions.
Although its position has indeed changed… really ask yourself how fast it moved?
What would it mean for the pixel to travel at different speeds across the screen without changing the size of the increment?
Imagine that X starts at coordinate 5 again, but it's now incrementing by 2 until it reaches the edge of the screen.
Notice that a physical difference existed, i.e., 1 unoccupied pixel existed in-between each incremented position, instead of 0 pixels.
Therefore, did the pixel move twice as fast during step 6, compared to during step 2?
If the answer for step 8 is “yes” – then just this single hypothetical pixel can indeed represent time.
If the answer for step 8 is “no” – then for time to exist for a single massive particle, it requires some other form of positional representation, such as the position of other massive particles, or the positional changes of virtual particles that occur as quantum behaviour.
Conclusion: A single entity representing time as hypothesised above – time exists purely as the size of incremental position change; jumps; popping out of spatial existence, then popping back into existence at a different position.